The standard method of signal processing uses representation of band-limited signals by Nyquist rate samples. Unique deterMination of band limited signals requires a sequence of Nyquist rate samples from the entire interval (-.infin.,.infin.). Moreover, such a sequence contains no redundant information, because by Nyquist's Theorem for arbitrary sequence of real numbers {a.sub.m }.sub.m=Z such that ##EQU1##
there exists a .pi. band-limited signal f(t) such that f(m)=a.sub.m. Of course, in practice operators act (essentially) on a sequence of approximations of the input signal obtained using a suitable moving window and setting the values of the signal to 0 at all sampling points outside of the support of the window.
Historically, numerical signal processing is older than the use of digital computers. For example, fast Fourier transforms (FFT) were first performed by hand. Also, early hardware for data acquisition (A/D conversion) and for numerical data processing was very limited in speed and capacity. In such circumstances it was important that the signal be represented without any redundancy and that the implementations of signal processing operators required minimal number of basic arithmetical operations. This detrined the direction of the development of signal processing algorithms, which resulted in an extremely elaborate and powerful signal processing paradigm centered around the "minimalist" representation of signals by the Nyquist rate samples. This paradigm is usually referred to as "signal processing based on harmonic analysis."
However, such "minimast" approach to signal representation, free of any redundancy, need not always be optimal. Specifically, the standard signal processing operators act on sequences of values of the input signal at consecutive (Nyquist rate spaced) sampling points which are within the support of an appropriate window. These values are stored in some form of a shift register and the output value is then obtaned either from these samples alone (FIR procedures) or from these samples together with the previously computed value of the signal (IIR procedures). Both types of procedures necessarily produce significant delays and/or phase-shifts in the output signal.
Moreover, although the present-day hardware is capable of accessing the information contained between the Nyquist rate sampling points, no approach based on harmonic analysis has been able to encode this information without producing troublesome proliferation of numerical data which must be stored. The concept of "Signal Processor With Local Signal Behavior" had introduced a method for using the information contained between the Nyquist rate sampling pointswithout producing troublesome proliferation of numerical data which must be stored, and without significantly increasing the computational complexity of the algorithms for subsequent processing of the data.
A need exists for signal representation and processing that does not necessarily produce delays and/or phase-shifts in the output signal. Also, a need exists for signal representation and processing that is not limited by the standard approaches. In particular, a need exists for making use of the information contained between Nyquist rate points without producing troublesome proliferation of numerical data which must be stored. Moreover, given the present-day hardware, a need exists to access this information without increasmg the computational complexity of the algorithms. The present invention provides methods for the above tasks significantly improving on those decribed in the invention "Signal Processor With Local Signal Behavior".